[Solutions à bruit nul des équations linéaires de transport : un exemple]
On considère un exemple unidimensionnel classique d'équation de transport linéaire sans unicité des solutions faibles. En présence d'une perturbation donnée par un bruit multiplicatif convenablement choisi, l'équation se révèle bien posée. On identifie les deux solutions de l'équation déterministe obtenues dans la limite ou le bruit s'annule. On prouve aussi que la solution de viscosité nulle existe et qu'elle est différente des deux autres.
We consider a classical one-dimensional example of linear transport equation without uniqueness of weak solutions. Under a suitable multiplicative noise perturbation, the equation is well posed. We identify the two solutions of the deterministic equation obtained in the zero-noise limit. In addition, we prove that the zero-viscosity solution exists and is different from them.
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@article{CRMATH_2009__347_13-14_753_0,
author = {Attanasio, Stefano and Flandoli, Franco},
title = {Zero-noise solutions of linear transport equations without uniqueness: an example},
journal = {Comptes Rendus. Math\'ematique},
pages = {753--756},
publisher = {Elsevier},
volume = {347},
number = {13-14},
year = {2009},
doi = {10.1016/j.crma.2009.04.027},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.027/}
}
TY - JOUR AU - Attanasio, Stefano AU - Flandoli, Franco TI - Zero-noise solutions of linear transport equations without uniqueness: an example JO - Comptes Rendus. Mathématique PY - 2009 SP - 753 EP - 756 VL - 347 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.04.027/ DO - 10.1016/j.crma.2009.04.027 LA - en ID - CRMATH_2009__347_13-14_753_0 ER -
%0 Journal Article %A Attanasio, Stefano %A Flandoli, Franco %T Zero-noise solutions of linear transport equations without uniqueness: an example %J Comptes Rendus. Mathématique %D 2009 %P 753-756 %V 347 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.04.027/ %R 10.1016/j.crma.2009.04.027 %G en %F CRMATH_2009__347_13-14_753_0
Attanasio, Stefano; Flandoli, Franco. Zero-noise solutions of linear transport equations without uniqueness: an example. Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 753-756. doi : 10.1016/j.crma.2009.04.027. http://www.numdam.org/articles/10.1016/j.crma.2009.04.027/
[1] Transport equation and Cauchy problem for BV vector fields, Invent. Math., Volume 158 (2004) no. 2, pp. 227-260
[2] Small random perturbations of Peano phenomena, Stochastics, Volume 6 (1982), pp. 279-292
[3] Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989) no. 3, pp. 511-547
[4] A note on generalized flows, Phys. D, Volume 183 (2003) no. 3–4, pp. 159-174
[5] Well-posedness of the transport equation by stochastic perturbation | arXiv
[6] Generalized calculus and SDEs with non-regular drift, Stochastics Stochastics Rep., Volume 72 (2002) no. 1–2, pp. 11-54
[7] A singular large deviations phenomenon, Ann. Inst. H. Poincaré Probab. Statist., Volume 37 (2001) no. 5, pp. 555-580
[8] Phénomène de Peano et grandes déviations [Large deviations for the Peano phenomenon], C. R. Acad. Sci. Paris, Sér. I Math., Volume 332 (2001) no. 11, pp. 1019-1024 (in French)
[9] Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996
[10] Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1991
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